In mathematics, a inequality is an expression that indicates that one quantity is greater or less than another.
Rules for solving inequalities
- First rule: Trichotomy Law: For two arbitrary real numbers a and b, one and only one of these relations a - b> 0 or - b <0 holds.
Verification: We must bear in mind that a very important property of real numbers is that they have an order; the order of the real numbers allows comparing two numbers and deciding which of them is greater or if both are equal. Now, for two real numbers a and b, consider the quantity a - b. Due to the property of real numbers, we have to:
a - b> 0 or a - b <0
a> boa <b
Second rule: Yes to
Verification: Let a <b, therefore b - a> 0. Then:
(b + c) - (a + c) = b + c - a - c = b - a> 0
a + c <b + c
(b - c) - (a - c) = b - c - a + c = b - a> 0
a - c <b - c
Third rule: If a <b and c> 0, then a ∙ c <b ∙ c and ÷ c <b ÷ c. That is, if we multiply or divide both members of the inequality by the same quantity, the meaning of it does not change.
Verification: Let a <b, therefore b - a> 0. Since yc> 0, we have:
(b - a) ∙ c> 0 → b ∙ c - a ∙ c> 0 → → b ∙ c> a ∙ c
Similarly, as yc> 0, we have that 1⁄c> 0, therefore:
(b - a) ∙ (1 / c)> 0 → b / c - a / c> 0 → → b / c> a / c
Fourth rule: If a <b and c <0, then - (a ∙ c)> - (b ∙ c) and - (a ÷ c)> - (b ÷ c). That is, if we multiply or divide both members of the inequality by the same negative quantity, the meaning of it changes.
Verification: Let a <b, therefore b - a> 0. As c <0, then -c> 0 and consequently:
(b - a) ∙ (-c)> 0 → - (b ∙ c) + a ∙ c> 0 → → (a ∙ c) <(b ∙ c)
Analogously As c <0, then -c> 0 and - (1⁄c)> 0, therefore:
(b - a) ∙ (-1 / c)> 0 → -b / c + a / c> 0 → → a / c <b / c
The above rules can be used to get more rules.
Rules for graphing inequalities
- Construct a table of the inequality equations.
- Represent the solutions on the coordinate plane.
- Draw a line that passes through each point. If the inequality is <or>, the line is a dashed line. If the inequality is ≤ or ≥, the line is a continuous line.
- Take a point and replace it on the inequality. If it is fulfilled, the solution is the region where it is located, otherwise the solution will be the other region.